Problem: The number of mosquitoes in Anchorage, Alaska (in millions of mosquitoes) as a function of rainfall (in centimeters) is modeled by: $m(x)=-x^2+14x$ What is the maximum possible number of mosquitoes?
The number of mosquitoes is modeled by a quadratic function, whose graph is a parabola. The maximum number of mosquitoes is reached at the vertex. So in order to find the maximum number of mosquitoes, we need to find the vertex's $y$ -coordinate. We will start by finding the vertex's $x$ -coordinate, and then plug that into $m(x)$. The vertex's $x$ -coordinate is the average of the two zeros, so let's find those first. $\begin{aligned} m(x)&=0 \\\\ -x^2+14x&=0 \\\\ x^2-14x&=0 \\\\ x(x-14)&=0 \\\\ \swarrow &\searrow \\\\ x=0\text{ or }&x-14=0 \\\\ x={0}\text{ or }&x={14} \end{aligned}$ Now let's take the zeros' average: $\dfrac{({0})+({14})}{2}=\dfrac{14}{2}= 7$ The vertex's $x$ -coordinate is $ 7$. Now let's find $m({7})$ : $\begin{aligned} m( 7)&=-( 7)^2+14( 7) \\\\ &=-49+98 \\\\ &=49 \end{aligned}$ In conclusion, the maximum number of mosquitoes is $49$ million.